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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 24th 2010

added the definition to cyclic homology

next the task is to write out the details for how under the identification of the Hochschild complex with functions on the derived loop space, the cyclic complex is the $S^1$-equivariant functions on the derived loop space.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeNov 24th 2010

Thanks for the new material. We should, of course, have it under an entry cyclic homology and not under cyclic cohomology, eventually, when both cases get described (in many cases, not only for associative algebras). But later...

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 24th 2010

Let’s keep both in the same entry. Otherwise there will be lots of duplication. I’ll add the discussion of cohomology later.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeNov 24th 2010
• (edited Nov 24th 2010)

OK, for some time I agree. But eventually the details for special cases like for discrete associative algebras (as opposed to say schemes, spectra etc.) should really be in different entries and the basic entry in my understanding could be about the general categorical nonsense of utilizing Connes’ cyclic modules to get cyclic (co)homology like one uses simplicial objects in other situations. The geometric picture about ${S}^1$ equivariant spaces may fit with also quite well with topological cyclic homology.

The general subject under the traditional heading of cyclic homology is much wider subject than the tradiotional subject of Hochschild homology.

There is an interesting approach which you may like, due Cortinas, using crystalline site with infinitesimal thickenings. I do not know if it was ever compared to the picture in Toen and in BenZvi-Nadler.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeNov 24th 2010
• (edited Nov 24th 2010)

There is a very interesting thing when one does the cyclic homology for algebras over cyclic operads. Then there are two kinds which fit into a version of Connes exact sequence which agree for associative algebras because the operad for associative algebras is self-dual so the two different versions coincide and the Connes- exact sequence has only Hochschild and cyclic terms. That is in Getzler-Kapranov paper.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeDec 2nd 2010

I added the definition of the relative cyclic homology into cyclic homology.

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeSep 5th 2013

New entry epicyclic space redirecting also epicyclic category and epicyclic set.

• CommentRowNumber8.
• CommentTimeJul 8th 2014

Some edits at cyclic homology, still under construction. Also changed the title from cyclic cohomology to cyclic homology.

• CommentRowNumber9.
• CommentTimeJul 8th 2014

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJun 27th 2021

• Jean-Louis Loday, Section 4 of: Free loop space and homology, Chapter 4 in” Janko Latchev, Alexandru Oancea (eds.): Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematics and Theoretical Physics 24, EMS 2015 (arXiv:1110.0405, ISBN:978-3-03719-153-8)

and made more explicit in the entry that the theorem in section 4 is reviewing a theorem due to Jones.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 27th 2021

also Cor. 7.3.1 in:

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeAug 18th 2021